The commons dilemma: Strategic common pool resource extraction behavior

The commons dilemma: Strategic common pool resource extraction behavior | Research Square window.SnipcartSettings = { analytics: { enabled: false } }; (function() { var accessVector = localStorage.getItem('access_vector') || ''; window.dataLayer = window.dataLayer || []; if (accessVector) { window.dataLayer.push({ user: { profile: { profileInfo: { snid: accessVector } } } }); } })(); (function(w,d,s,l,i){w[l]=w[l]||[];w[l].push({'gtm.start':new Date().getTime(),event:'gtm.js'});var f=d.getElementsByTagName(s)[0],j=d.createElement(s),dl=l!='dataLayer'?'&l='+l:'';j.async=true;j.src='https://www.googletagmanager.com/gtm.js?id='+i+dl;f.parentNode.insertBefore(j,f);})(window,document,'script','dataLayer','GTM-K279D39R'); Browse Preprints In Review Journals COVID-19 Preprints AJE Video Bytes Research Tools Research Promotion AJE Professional Editing AJE Rubriq About Preprint Platform In Review Editorial Policies Our Team Advisory Board Help Center Sign In Submit a Preprint Cite Share Download PDF Research Article The commons dilemma: Strategic common pool resource extraction behavior Haile Tewele Berhe This is a preprint; it has not been peer reviewed by a journal. https://doi.org/ 10.21203/rs.3.rs-2654790/v1 This work is licensed under a CC BY 4.0 License Status: Posted Version 1 posted You are reading this latest preprint version Abstract The paper presents the strategic decision of common pool resource users at local level. Common pool resources typically characterized by the difficulty to exclude appropriation by others and the high level of rivalry (subtract ability). In this study, an application of game theory has been adopted to show individual resource use behavior. A game that better captures this underlying mechanism is the common pool resource game. Common pool resources with unlimited access are generally characterized by the existence of negative externality. Assuming a symmetric users’ payoff function, static analysis for a simple mathematical model using solution method for strategic common pool game come up with extraction effort levels exceed the socially optimal level at equilibrium. This implies that unregulated common pool resource game poses social dilemma; when players interact only once or finitely repeated. On the contrary, the infinitely repeated framework shows cooperation (socially optimal level of extraction) may sustain by a self-enforcing agreement provided individuals are sufficiently patient. This frame work indicates that cooperation may sustain for a long period of time, if the initial gain from deviation is small or losses from counter threat after deviation is large. Common pool resource Game Extraction effort Self-enforcing cooperation Figures Figure 1 1. Introduction 1.1. Natural resource extraction: users’ strategic decision The most typical causes of natural resource problems the world currently confronted with, including depletion of fisheries, forests, irrigation systems, grazing lands, minerals, fresh water, and biodiversity loss are economic and determined by institutions including the legal system, cultural norms, market structures, and other rules of behavior affecting decision-making incentives. Even if access to resources is limited to a specific group of individuals or a community, socially excessive resource use may occur if appropriation externalities are present, that is, if increased resource extractions by one user/group of users reduce the net yield obtained by other users either instantaneously or over time [ 1 , 2 ]. In this situation, each individual resource-user ignores the costs he/she imposes on other resource-users, and hence, from a social welfare point of view, puts too much effort into resource harvesting. As all individuals face the same situation, the resulting resource stock comes to be smaller than the one that maximizes aggregate payoff. The combination of appropriation externalities and lack of individualized and sufficiently well-defined property rights provides a classic case for government intervention, but socially optimal resource management may also be achieved by means of cooperation among resource-users. 1.2. Common-pool resources The commons or common-pool resources are resources that are difficult to exclude (i.e., it is difficult to limit their appropriation) and have high levels of subtract ability (i.e., their use or appropriation by a person or a group of people reduces the ability of others to benefit from them). When resource extraction is rival tragedy of the common follows. In essence, the tragedy of the commons arises because of an externality. Environmental degradation is a modern tragedy of the commons. When one or more individuals appropriate the common resource, it reduces the quality or quantity of the resource available for others. Examples are irrigation systems, fishers, ground water basins, grazing lands, and some forest resources. For open access common pool resources (CPR), it is hard for such resources to allocate equitably nor optimally. The remedy for this particular externality must instead be determined by the individuals affected by it, through negotiation. Users of such resources may in many instances overcome incentives to destroy the resources by developing rules that enable them utilize these resources effectively [ 2 ]. Potentially, there are other ways of improving the use of the resources. It would be feasible for the users that share a resource to merge. Cooperation can help in managing local (intra-national) resource problems. That is, as local communities often are homogeneous (composed of people who live near each other and who have intimate connections, shared values, and common histories) it is much easier to cooperate. Government may assign resource rights to the community group, and thereby prevent entry by outsiders. Of course, free riding may make effective management of common property resources harder. Some recent studies focusing on individual natural resource use behavior have tried to adopt game theory. Most studies for example, [ 3 , 4 , 5 , 6 , 7 , 8 , 9 ] have focused on public goods (PG) games, and analyzed the extent to which especially punishments and rewards affect behavior. This study focuses on environmental problems, especially, those related to (renewable) natural resource use, in which individual’s harvesting activity negatively affects the returns to all other users of the resource under consideration. A game that better captures this underlying mechanism is the common pool resource (CPR) game. While in some cases users may construct formal or in formal institutions to more efficiently manage the commons. Generally such unexclusive resources are susceptible for over-consumption which referred as, the “tragedy of the commons”. This paper briefly describes the basis for common pool resource game models compatible with the real-world environmental problems; show mathematically how the individual level of extraction effort differs from the socially optimal. Provide a cooperation framework in which individual’s choose the social welfare maximizing level of extraction effort over the unilateral payoff maximizing level. And argue the simplest framework used for the economic analysis of CPR. 2. Set Up Of The Common Pool Resource Game The basis for an open access natural resource extraction oriented game is the finitely repeated common pool resource game similar to that of [10⦌. Let there is a closed community of N resource users, where N > 1 and the community members have unrestricted access to the common pool resource. In every period t = 1, …, T , each user i = 1,. . ., N can allocate a fixed endowment of effort, e > 0 , between CPR extraction and an alternative economic activity (the outside option). Assuming extraction of CPR by individuals who have access to the open access resource increases in effort level, where extraction effort exerted by user i in period t is denoted by x i,t , and hence user i ’s effort devoted to the outside option equals ( e-x i,t ). The outside option yields a fixed per-unit wage rate, w. When exerting extraction effort, users incur cost that is linear in extraction effort. The marginal cost is assumed to be constant and equal to c. In addition to the above assumptions, this paper assumes complete information throughout. Assume again the group’s benefit in period t , B t , depends on the aggregate amount of extraction effort in that period, \({X}_{t }=\sum _{i=1}^{N}{x}_{i,t}\) , where price is assumed constant and equal to 1. Let the benefit function is given by \({B}_{t}\left({X}_{t}\right)\) = \({ aX}_{t}\) - \({bX}_{t}^{2}\) , for any positive constants a and b. User i ’s share in these benefits is proportional to his/her share in aggregate extraction effort ( \(\frac{{x}_{i,t}}{{X}_{t}}\) ). Hence, user i ’s aggregate payoff, 𝛱 i,t in period t equals: $${\varPi }_{i,t}\left({x}_{i,t},{X}_{t}\right)=w\left(e-{x}_{i,t}\right)+\frac{{x}_{i,t}}{{X}_{t}} \left({aX}_{t}-{bX}_{t}^{2}\right)-{cx}_{i,t} \left(1\right)$$ For the sake of simplicity, let’s limit the common pool resource extraction game to the one shot game. And the socially optimal extraction effort is the one that maximizes the unweighted payoffs of all N users in the group defined in Eq. (1). In which the solution concept refers to the sort of cooperative games that reflect actions taken in a coordinated way by all parties. Proposition 1 For unregulated open access common pool resource game, every user i in the community chooses an equilibrium level of harvesting effort that exceeds the social optimum. Proof In order to show whether the equitable socially optimal extraction effort level is below the individual unilateral equilibrium level of effort, first it is important to specify the net benefit (payoff) of all N - users in the group: $$\varPi \left({X}_{t}\right)=w\left(Ne-{X}_{t}\right)+{aX}_{t}-{bX}_{t}^{2} -{cX}_{t} \left(2\right)$$ We can apply the first order derivative on Eq. (2) to obtain the aggregate optimal extraction level of effort. $$\frac{\partial \varPi \left({X}_{t}\right) }{\partial {X}_{t}}=\frac{\partial \left[\left({Ne-X}_{t}\right)w+{aX}_{t}-{bX}_{t}^{2} -{cX}_{t}\right]}{\partial {X}_{t}}=0$$ Hence, the optimal extraction effort level, \({X}_{t}=\) \({X}^{*}=\frac{a-c-w }{2b}\) , for \(a-c-w>0\) . And the individual equitable socially optimal extraction effort level is: \({{x}_{i,t}=x}^{*}=\frac{{X}^{*}}{N}=\frac{a-c-w }{2Nb}\) (a) Let us use \({x}_{0}\) to denote common pool resource game in the absence of peer regulation. Assuming that individuals are rational and aim to maximize their own payoffs, user i’s best response function to any level of aggregate extraction effort by all others ( \({X}_{-i,t}=\sum _{j\ne i}^{N}{x}_{j,t}\) ) is obtained by rearranging Eq. (1) and apply first order condition (foc). $${\varPi }_{i,t}\left({x}_{i,t},{X}_{t}\right)=w\left(e-{x}_{i,t}\right)+\frac{{x}_{i,t}}{{X}_{t}} \left({aX}_{t}-{bX}_{t}^{2}\right)-{cx}_{i,t}$$ \(= w\left(e-{x}_{i,t}\right)+{x}_{i,t} \left(a-b{X}_{t}\right)-c{x}_{i,t}\) , substituting \({X}_{t}={x}_{i,t}+{X}_{-i,t}\) $${\varPi }_{i,t}\left({x}_{i,t},{X}_{-i,t}\right)=w\left(e-{x}_{i,t}\right)+{ax}_{i,t}-{bx}_{i,t}^{2} -b{x}_{i,t} {X}_{-i,t}-c{x}_{i,t}$$ Let’s apply foc with respect to \({x}_{i,t}\) on this version of Eq. (1): \(\frac{\partial {\varPi }_{i,t}\left({x}_{i,t}, {X}_{-i,t}\right) }{\partial {x}_{i,t}}=\frac{\partial w\left(e -{ x}_{i,t}\right)+{ ax}_{i,t}-{ bx}_{i,t}^{2} - b{x}_{i,t} {X}_{-i,t}- c{x}_{i,t}}{\partial {x}_{i,t} }\) = 0 \(=a-w-2b{x}_{i,t}-b{X}_{-i,t}-c=0\) , and \({ x}_{i,t}\left({X}_{-i,t}\right)=\frac{a - c - w}{2b}-\frac{{X}_{-i,t}}{2}\) (b) For a symmetric individual extraction effort, \({ x}_{i}\) we have \({ x}_{i}={x}_{j}=x\) and \({X}_{-i,t}=\sum _{j\ne i}^{N}{x}_{j,t }=(\text{N}-1){x}_{i,t}\) . Substituting \({X}_{-i,t}=(\text{N}-1){x}_{i,t}\) in the first order condition equation (b) above we obtain: $${ x}_{i,t}=\frac{a - c - w}{2b}-\frac{{\left(N- 1\right)x}_{i,t}}{2}$$ $${ x}_{i,t} +\frac{{\left(N-1\right)x}_{i,t}}{2}=\frac{a - c - w}{2b}$$ $${ x}_{i,t}\left(1+\frac{N-1}{2}\right)=\frac{a - c - w}{2b}$$ Thus \({ x}_{i,t}={x}_{j,t}={x}_{0}^{NE}=\) \(\frac{a - c - w}{b(N+1)}\) (c) From equations (a) and (b) above compare the socially optimal level of extraction effort ( \({x}^{*})\) for individuals in the group with the unique symmetric Nash equilibrium extraction effort \({x}_{0}^{NE}\) , \({x}_{0}^{NE}\) > \({ x}^{*}\) for all N > 1 , and \(a-c-w>0\) . This implies that the unregulated common pool resource game poses a social dilemma. Social dilemma indicates for the situation in which individuals’ and society’s interest is in conflict. In the real world, for open access resources for e.g., forest, fishery, fresh water, grazing land and mining when the number of users for the unrestricted resource increased, every additional extraction by new users increases the amount of time and effort required to extract extra resource. In Fig. 1 under, social benefits and costs of forest extraction (e.g., fuel wood, charcoal or grass) for the benefit function specified above are illustrated. Total benefits of extraction are calculated by multiplying price per unit by the amount of harvest. The marginal benefit curve is down ward sloping because the greater the amount of extraction effort expended, the smaller the resulting remaining resource will be. And the smaller left over resource, the smaller harvest per unit of effort expended. The optimal level of extraction effort level, \({x}^{*}\) in Fig. 1 (b) is the level where, the marginal benefit would be just equal to the marginal social cost, implying that net benefits are maximized. The marginal social cost includes not only the private operating costs but also the social cost of depleting the open access resource. With all extractors having completely unrestricted access to the resource the resulting allocation would be above the optimal level. As Fig. 1 (a) shows, individuals would exploit the resource until their total benefit equaled total cost; or equivalently as Fig. 1 (b) shows until their marginal benefit equals private marginal cost, implied by the level of effort \({ x}_{0}\) . Excessive exploitation of the resource occurs because individuals have no incentive to leave resource after. In the presence of sufficient demand for the common pool resource, unrestricted access will cause resource to be over exploited. This is because the individuals/groups exploiting the open access would not have any incentive to conserve as the benefits derived from restraint would, to some extent, be captured by other extractors. Hence, it is essential for the group who have unlimited resource access to look for a mechanism that help in addressing the social dilemma in common pool resources. The dilemma is characterized by the sub-optimal equilibrium of non-cooperative game associated with an individual’s decision that only satisfies their own selfish interest. Thus, devising an arrangement that may enforce users select the socially optimal level of effort instead of the business as usual or (unregulated) effort level is suggested in our framework. That is, the arrangement is required to encompass a kind of agreement that motivate individuals to self-select the optimal (cooperative) level of extraction effort over the business as usual effort level. For a static common-pool resource, every player (user) extracts too much and leaves smaller pie for the other users. If we extend the static CPR extraction game in to a simple two stage dynamic common-pool problem, efficiency may further reduced. Every user may be incentivized to extract a lot at the first stage and this will discourage other from extracting at the second stage. The outcome is that even more of the resource may be extracted at the beginning. Hence, unless the users introduced some regulatory framework and act accordingly, intertemporal allocation of unregulated common pool resource becomes more inefficient: too much is extracted in the beginning and too little later on (conditional on the total amount extracted in the beginning). 3. The Infinitely Repeated Common Pool Resource Extraction Game For static common pool resource game users make extraction effort choice once simultaneously. In reality as individual user live together and reap resource benefit for longer period of time, strategic decisions are required to take this in to account. In contrast to the situation where agents interact only once, any mutually beneficial outcome can be sustained as equilibrium when agents interact repeatedly and frequently [11⦌. Formally, repeated games refer to a class of models where the same set of agents repeatedly plays the same game, called the ‘stage game’, over a long time horizon. Behavioral studies indicate that the equilibrium strategy for a finitely repeated games or games with certain number of rounds is the same as the equilibrium of the one shot game. In the absence of binding agreements, infinitely repeated interactions allow for socially beneficial outcomes. That is, if interactions are infinitely repeated, then socially beneficial outcomes that cannot be sustained by players with short-term objectives may sustained by players with long-term objectives. Although one may argue that players do not really live infinitely long (so that the finite horizon case is more realistic), there is a good reason to consider the infinite horizon models. Even though the time horizon is finite, if players do not know in advance exactly when the game ends, the situation can be formulated as an infinitely repeated game. In a general game, any outcome which Pareto dominates the Nash equilibrium can be sustained by a strategy which reverts to the Nash equilibrium after a deviation. Such a strategy is called a trigger strategy. And the strategies with payoff sets Pareto dominate the Nash equilibrium may ascertain by the Folk theorem. The Folk theorem asserts that when players are equally and sufficiently patient, all payoffs that are socially feasible and Pareto dominate the minimax are subgame perfect equilibrium payoffs. Alternatively the Folk theorem may be defined as: for some discount factor sufficiently close to unity, any set of feasible payoffs that are preferred by all individuals to the Nash equilibrium payoffs can be supported as subgame perfect equilibria for the repeated game. Hence, for infinitely repeated game, cooperation may sustain through a self-enforcing agreements provided that individuals are sufficiently patient. This is in line with Barrett [ 12 ] who suggested the possibility of enforcing cooperation for international environmental problems. The repeated game differs from its one-shot version that individuals who deviate from a promise to play cooperation can be punished by members of the group. Suppose that the game is played forever and discount rates are vanishingly small. Suppose, moreover, that the players play the so-called “Triger” strategy- each begins by playing low effort. Thereafter, each player plays low effort provided no player played high effort in the past. Otherwise, each individual in the group apply high effort. 3.1. Infinitely repeated common pool resource extraction stage game Let the common pool resource game is a repeated game consists of a stage game and a set of times when the stage game is played. Assume, there are N- players (users) indexed by \(i or j\in N=\left\{1,\dots ,n\right\}.\) Assume again each individual writes agreement with at most one group of players, the coalition to which he/she belongs at stage 1. For the sake of simplicity, we assume only two possible extraction efforts \({x}_{i}\in \{ \underset{\_}{x} ,\stackrel{-}{x}\}\) . Where, \(\underset{\_}{x}\) and \(\stackrel{-}{x}\) stands for relatively low and high extraction efforts respectively. Assume again at the harvesting stage (stage 2) users who have access to the resource simultaneously decide between more or less effort, \({x}_{i}\in \{ \underset{\_}{x} ,\stackrel{-}{x}\}\) . Let in the unregulated harvesting effort individuals decision constitute a prisoners’ dilemma game. That is, individuals are benefited from more own harvesting effort for any fixed harvesting effort level from all other players, \({X}_{-i}=\sum _{j\ne i}^{N}{x}_{j,}\) but every individual would be better off if everyone else have less instead of more effort. The prisoners’ dilemma game is a reasonable stage game in this game of extraction effort for common pool resources. If \({B(x}_{i})\) and \(c\) are the benefit of extraction effort and marginal cost of extraction efforts respectively, net payoff from extraction is given by: $${\varPi }_{i}\left({x}_{i}\right)=B\left({x}_{i}\right)-c\sum _{i=1}^{N}{x}_{i}$$ 3 Hence for \({x}_{i}\in \{ \underset{\_}{x} ,\stackrel{-}{x}\}\) , less extraction effort, \(x=\underset{\_}{x}\) will simply be the individually optimal extraction effort if the one period net payoff from less extraction effort outweighs that of more extraction effort. \(B\left({\stackrel{-}{x}}_{i}\right)-c\sum _{i=1}^{N}{\stackrel{-}{x}}_{i}\) < \(B\left({\underset{\_}{x}}_{i}\right)-c\sum _{i=1}^{N}{\underset{\_}{x}}_{i}\) Assuming all individuals in the group exert same extraction effort at optimality, the above equation may be written as: \(B\left({\stackrel{-}{x}}_{i}\right)-cN\stackrel{-}{x}\) < \(B\left({\underset{\_}{x}}_{i}\right)-cN\underset{\_}{x}\) $$B\left({\stackrel{-}{x}}_{i}\right)- B\left({\underset{\_}{x}}_{i}\right)<cN(\stackrel{-}{x}-\underset{\_}{x})$$ \(\frac{B\left({\stackrel{-}{x}}_{i}\right)- B\left({ \underset{\_}{x}}_{i}\right)}{c(\stackrel{-}{x}-\underset{\_}{x}) } <N\) (d) Similarly, high extraction effort \(x=\stackrel{-}{x}\) will be a dominant strategy if the net payoff from more extraction effort exceeds that of less extraction effort, taking in to account only individuals own extraction cost. \(B\left({\stackrel{-}{x}}_{i}\right)-c{\overline{x}}_{i}\) > \(B\left({\underset{\_}{x}}_{i}\right)-\) c \({\underset{\_}{x}}_{i}\) \(B\left({\stackrel{-}{x}}_{i}\right)-\) \(B\left({\underset{\_}{x}}_{i}\right)\) > \(c{\overline{x}}_{i}-\) c \({\underset{\_}{x}}_{i}\) \(\frac{B\left({\stackrel{-}{x}}_{i}\right)- B\left({\underset{\_}{x}}_{i}\right)}{c({\stackrel{-}{x}}_{i}- {\underset{\_}{x}}_{i})}>1\) (e) Thus, the common pool resource extraction effort game is a prisoner’s dilemma game if both (d) and (e) holds true. And from equations (d) and (e) we reach at the expression under. $$1<\frac{B\left({\stackrel{-}{x}}_{i}\right)- B\left({\underset{\_}{x}}_{i}\right)}{c({\stackrel{-}{x}}_{i}- {\underset{\_}{x}}_{i})}<N$$ 4 If the users agree to cooperate they can reach a better equilibrium than the business as usual equilibrium that happens in the Prisoner’s dilemma game. In order for cooperation \(x=\underset{\_}{x}\) sustain as a Pareto optimal subgame perfect equilibrium, the worst equilibrium, \(x=\stackrel{-}{x}\) might be used as threat to enforce better equilibrium. That is, \(x=\stackrel{-}{x}\) can be sustain as a trigger strategy in a subgame perfect equilibrium (SPE), where any deviation requires the players to revert to the business as usual equilibrium forever. This is in line to Barrett [12⦌ where he suggested a strategy of reciprocity to enforce compliance for international environmental agreements. Proposition 2 Cooperation hardly sustains if: i. Individuals are heavily discounting the value of cooperating in the future (i.e., 𝛿 is small). ii. the number of individuals in the group is relatively few Proof for proposition 2 (i): For an infinitely repeated game with discount rate, r and discount factor, \(\delta =\frac{1}{1+r}\) , in the absence of binding agreements either due to the absence of third party to enforce agreements or because enforcing agreement by a third party is costly, self-enforcing agreement may sustain if the discounted payoff streams from cooperation ( \({V}_{i}^{c}\) ) out weight the payoff streams from breaking cooperation ( \({V}_{i}^{d}\) ). Where, the discounted payoff stream for individual i is, \({V}_{i}=\sum _{t=0}^{\infty }{\delta }^{t-1}{\varPi }_{it}\) , for 0 < 𝛿 < 1 \({V}_{i}^{c}=\frac{B\left(\underset{\_}{x}\right) - cN\underset{\_}{x}}{(1 - \delta )}= ; {V}_{i}^{d}=\text{B}\left(\stackrel{-}{x}\right)-c\stackrel{-}{x}-c\left(N-1\right)\underset{\_}{x}+\frac{\delta [\text{B}\left(\stackrel{-}{x}\right) - cN\stackrel{-}{x})] }{(1 - \delta )}\) , mathematically this may be specified as: \({V}_{i}^{c}\ge\) \({V}_{i}^{d}\) $$\frac{B\left(\underset{\_}{x}\right) - cN\underset{\_}{x}}{(1 - \delta )}\ge \text{B}\left(\stackrel{-}{x}\right)-c\stackrel{-}{x}-c\left(N-1\right)\underset{\_}{x}+\frac{\delta [\text{B}\left(\stackrel{-}{x}\right) - cN\stackrel{-}{x})] }{(1 - \delta )}$$ $$\frac{B\left(\underset{\_}{x}\right) - cN\underset{\_}{x}}{(1 - \delta )}\ge \text{B}\left(\stackrel{-}{x}\right)-c\left[\stackrel{-}{x}+\left(N-1\right)\underset{\_}{x}\right]+\frac{\delta [\text{B}\left(\stackrel{-}{x}\right) - cN\stackrel{-}{x})] }{(1 - \delta )}$$ \(\text{B}\left(\stackrel{-}{x}\right)-B(\underset{\_}{x}\) ) \(\le \text{c}\left(\stackrel{-}{x} -\underset{\_}{x}\right)\left[\delta N+\left(1-\delta \right)\right]\) \({\delta }\ge\) \(\widehat{\delta }\equiv \frac{1}{N-1}\left[\frac{\text{B}\left(\stackrel{-}{x}\right) - B\left(\underset{\_}{x}\right)}{\text{c}\left(\stackrel{-}{x} - \underset{\_}{ x}\right)}-1\right]<1\) let \(\frac{\text{B}\left(\stackrel{-}{x}\right) - B\left(\underset{\_}{x}\right)}{\text{c}\left(\stackrel{-}{x} - \underset{\_}{x}\right)}=M,\) where \(1<M<N\) \({\delta }\ge\) \(\widehat{\delta }\equiv \frac{M - 1}{N - 1}<1\) This expression indicates in order to sustain cooperation, less extraction effort \(x=\underset{\_}{x}\) must be a SPE and the discount factor needs to be sufficiently high. For \({\delta }<\) \(\widehat{\delta }\) (a critical discount factor), because of the free riding incentive, the business as usual \(x=\stackrel{-}{x}\) will be an equilibrium extraction effort. However, if individuals are patient or very far sighted they will not easily deviate from the cooperation and temptation to free ride could be small. In addition to having large mental horizon, cooperation can be sustained with very little initial gain from deviation, large loss from counter threat or with short detection time for compliance after the game starts. However, where resource users are involved in a repeated game and trigger strategies are used to sustain cooperation, individuals may cooperate and choose an effort \(x=\underset{\_}{x}< \stackrel{-}{x}\) . The trigger strategy specifies that each party conforms to the agreement in each stage t as long as the parties have complied in all \(t-1\) prior stages; otherwise the parties deviate by breaking the arrangement. Proof for proposition 2 (ii): $$\frac{\partial {V}_{i}^{c} - { V}_{i}^{d})}{\partial N}=\frac{\partial }{\partial N}\left\{\frac{B\left(\underset{\_}{x}\right) - cN\underset{\_}{x}}{(1 - \delta )}\right\}-\frac{\partial }{\partial N}\left\{\text{B}\left(\stackrel{-}{x}\right)-c\stackrel{-}{x}-c\left(N-1\right)\underset{\_}{x}+\frac{\delta [\text{B}\left(\stackrel{-}{x}\right) - cN\stackrel{-}{x})] }{\left(1 - \delta \right)}\right\}$$ $$=c\underset{\_}{x}- \frac{c\underset{\_}{x}}{1 - \delta }+\frac{\delta c\stackrel{-}{x}}{1 - \delta }=c\underset{\_}{x}-\frac{c\underset{\_}{x}}{1 - \delta }+\frac{\delta c\stackrel{-}{x}}{1 - \delta }$$ \(=\) \(\frac{\delta }{1 - \delta }c\stackrel{-}{x}-\frac{\delta }{1 - \delta }c\underset{\_}{x}\) \(\frac{\partial {V}_{i}^{c} -{ V}_{i}^{d})}{\partial N}\) \(=\left(\stackrel{-}{x}-\underset{\_}{x}\right)\frac{\delta }{1 - \delta }c>0\) ; For \(0<\delta <1\) This indicates as long as group members are able to monitor sufficiently each other, motivating cooperation will be easier if the group size is relatively large. On the contrary, cooperation hardly sustains when number of individuals in the group is relatively few (i.e., N is small), or if individuals are heavily discount the value of cooperating in the future (i.e., 𝛿 is small). Other studies for e.g., [8⦌ concluded that informal sanctions like social pressure and disapproval of decisions may increase cooperation for public good game. Although in this paper the compliance constraint is limited to the condition in which only one of the individuals in the group deviate from the agreed level, the result can be generalized to the condition where s < N , individuals deviate from cooperation and as a result the game revert to business as usual then after. A simple common pool resource game has used in this manuscript, but this may open for extended future research along these lines. 4. Summary This paper presents both the finitely game (in its one time version) and infinitely repeated dynamic game formulations in which N members of a users’ group apply extraction effort that maximizes payoff from a common pool resource. For the finitely repeated move game, individual users are assumed to be homogeneous. And the payoff-functions are assumed to be concave and symmetric. Hence unconstrained optimizations are applied to find equilibrium solutions easily. The one time game of the finitely repeated framework clearly indicates the individual unregulated (business as usual) harvesting effort exceeds the social welfare maximizing level. Typically, this indicates the existence of negative externality in an open access common pool resources. An expression for infinitely repeated resource extraction game, where players have only two strategies (low and high extraction efforts) and played prisoners’ dilemma as an individual payoff maximizing at equilibrium has also been analyzed. This formulation entails users may form a users’ group and design an arrangement that enables them to self-select cooperation over the business as usual equilibrium that happens in the Prisoner’s dilemma game. In this formulation of the game, in order for low extraction effort sustain as equilibrium, the business as usual equilibrium \(x=\stackrel{-}{x}\) is used as a threat (trigger strategy) for any deviations from cooperation. Besides, the strategies chosen by each player are assumed to be observable. Hence, this study has revealed, for the infinitely repeated game formulation, the socially optimal extraction effort may sustain through a self-enforcing agreement provided that individuals are sufficiently patient. In other words, cooperation may sustain over a long period of time if the initial gain from deviation is small or losses from counter threat after deviation is large. Meanwhile, initial gain from deviation becomes small if the time of detection for compliance after the game starts is short and the number individuals in the group are relatively large. And losses from counter threat (reverting in to business as usual equilibrium) is large, if the group size is large, or if individuals are very patient (i.e, the common discount factor, 𝛿 is large). Declarations Ethical approval Not applicable Competing interests I have assured that there is no potential financial or non-financial competing interests from others related with this manuscript. Authors’ contribution The corresponding author (HTB) has carried out the study beginning from problem identification, design and writing the manuscript titled the commons dilemma: Strategic common pool resource extraction behavior. Besides, HTB has approved for publication in the Environmental Modeling and Assessment. Funding No funds or other support were received during the preparation of this manuscript. Availability of data and materials The materials used for this study can be provided on request. References Dell’Angelo, J., D’odorico, P., Rulli, M. C., & Marchand, P. (2017). The tragedy of the grabbed commons: Coercion and dispossession in the global land rush. World Development , 92 , 1-12. Ostrom, E., Gardner, R., Walker, J., & Walker, J. (1994). Rules, games, and common-pool resources . University of Michigan press. Kamijo, Y., Nihonsugi, T., Takeuchi, A., & Funaki, Y. (2014). Sustaining cooperation in social dilemmas: Comparison of centralized punishment institutions. Games and Economic Behavior , 84 , 180-195. Anderson, C. M., & Putterman, L. (2006). Do non-strategic sanctions obey the law of demand? The demand for punishment in the voluntary contribution mechanism. Games and Economic Behavior , 54 (1), 1-24. Carpenter, J. P. (2007). Punishing free-riders: How group size affects mutual monitoring and the provision of public goods. Games and Economic Behavior, 60(1), 31-51. Fehr, E., & Gächter, S. (2000). Cooperation and punishment in public goods experiments. American Economic Review , 90 (4), 980-994. Nikiforakis, N. (2008). Punishment and counter-punishment in public good games: Can we really govern ourselves?. Journal of Public Economics , 92 (1-2), 91-112. Noussair, C., & Tucker, S. (2005). Combining monetary and social sanctions to promote cooperation. Economic Inquiry , 43 (3), 649-660. Swope, K. J. (2002). An experimental investigation of excludable public goods. Experimental Economics , 5 , 209-222. Ostrom, E., Walker, J., & Gardner, R. (1992). Covenants with and without a sword: Self-governance is possible. American political science Review, 86(2), 404-417. Kandori, M. (2006). Repeated Games, Entry in The New Palgrave Dictionary of Economics (No. CIRJE-F-395). CIRJE, Faculty of Economics, University of Tokyo. Barrett, S. (2005). The theory of international environmental agreements, Sais, Johns Hopkins University, Rome Building, 1619 Massachusetts Avenue, NW Washington, DC 20036-2213, USA. Additional Declarations No competing interests reported. Cite Share Download PDF Status: Posted Version 1 posted You are reading this latest preprint version Research Square lets you share your work early, gain feedback from the community, and start making changes to your manuscript prior to peer review in a journal. As a division of Research Square Company, we’re committed to making research communication faster, fairer, and more useful. We do this by developing innovative software and high quality services for the global research community. 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Also discoverable on Platform About Our Team In Review Editorial Policies Advisory Board Help Center Resources Author Services Accessibility API Access RSS feed Manage Cookie Preferences © Research Square 2026 | ISSN 2693-5015 (online) Privacy Policy Terms of Service Do Not Sell My Personal Information {"props":{"pageProps":{"initialData":{"identity":"rs-2654790","acceptedTermsAndConditions":true,"allowDirectSubmit":true,"archivedVersions":[],"articleType":"Research Article","associatedPublications":[],"authors":[{"id":182882386,"identity":"59c6749f-bb3d-4be5-be2b-078ce40ab3a1","order_by":0,"name":"Haile Tewele Berhe","email":"data:image/png;base64,iVBORw0KGgoAAAANSUhEUgAAAZAAAAAyAQMAAABI0h/eAAAABlBMVEX///8AAABVwtN+AAAACXBIWXMAAA7EAAAOxAGVKw4bAAAA8UlEQVRIiWNgGAWjYBACAwbGxgNgFjsD4wMgxcNHhJaGAwwJQBYzA7MBSAsbYS0MDDAtbBIgEYJazNkPNxz4+MMmsZ+Z+Vjl1xw7GTYG5oePbuDRYtmT2HBwRkJa4sxmtrTbstuSgQ5jMzbOweewA4kNh3kSDuduOMxjdltyGzNQCw+bNF4t5x82HP6T8D93/2H+b8WS2+qJ0HIDaAtDwoHcDUDzGT9uO0yMlocNB3vSkutnHGYzlmbcdpyHjZmQX86nP3zww8bOmL+9+eHHn9uq7fnZmx8+xqcFBTDzgElilYMA4w9SVI+CUTAKRsGIAQA7mEvj3G1UKQAAAABJRU5ErkJggg==","orcid":"","institution":"Mekelle University","correspondingAuthor":true,"submittingAuthor":false,"prefix":"","firstName":"Haile","middleName":"Tewele","lastName":"Berhe","suffix":""}],"badges":[],"createdAt":"2023-03-04 12:14:18","currentVersionCode":1,"declarations":"","doi":"10.21203/rs.3.rs-2654790/v1","doiUrl":"https://doi.org/10.21203/rs.3.rs-2654790/v1","draftVersion":[],"editorialEvents":[],"editorialNote":"","failedWorkflow":false,"files":[{"id":34300966,"identity":"a45784b7-0fcd-4841-8b21-3cb9a7ba8dfa","added_by":"auto","created_at":"2023-03-15 14:38:46","extension":"png","order_by":1,"title":"Figure 1","display":"","copyAsset":false,"role":"figure","size":38620,"visible":true,"origin":"","legend":"\u003cp\u003eGraphical representation for open access resource extraction (a) using total benefit and total costs (b) using marginal benefit and marginal costs\u003c/p\u003e","description":"","filename":"1.png","url":"https://assets-eu.researchsquare.com/files/rs-2654790/v1/a409be7cb87bdc3f2f94351b.png"},{"id":35975453,"identity":"24993373-6e1e-4441-9f4d-285b268927c2","added_by":"auto","created_at":"2023-04-19 04:29:30","extension":"pdf","order_by":0,"title":"","display":"","copyAsset":false,"role":"manuscript-pdf","size":312606,"visible":true,"origin":"","legend":"","description":"","filename":"manuscript.pdf","url":"https://assets-eu.researchsquare.com/files/rs-2654790/v1/f759c71f-9745-4b62-97a2-95b11be2a9df.pdf"}],"financialInterests":"No competing interests reported.","formattedTitle":"The commons dilemma: Strategic common pool resource extraction behavior","fulltext":[{"header":"1. Introduction","content":"\u003cdiv id=\"Sec2\" class=\"Section2\"\u003e \u003ch2\u003e1.1. Natural resource extraction: users\u0026rsquo; strategic decision\u003c/h2\u003e \u003cp\u003eThe most typical causes of natural resource problems the world currently confronted with, including depletion of fisheries, forests, irrigation systems, grazing lands, minerals, fresh water, and biodiversity loss are economic and determined by institutions including the legal system, cultural norms, market structures, and other rules of behavior affecting decision-making incentives. Even if access to resources is limited to a specific group of individuals or a community, socially excessive resource use may occur if appropriation externalities are present, that is, if increased resource extractions by one user/group of users reduce the net yield obtained by other users either instantaneously or over time [\u003cspan citationid=\"CR1\" class=\"CitationRef\"\u003e1\u003c/span\u003e, \u003cspan citationid=\"CR2\" class=\"CitationRef\"\u003e2\u003c/span\u003e]. In this situation, each individual resource-user ignores the costs he/she imposes on other resource-users, and hence, from a social welfare point of view, puts too much effort into resource harvesting. As all individuals face the same situation, the resulting resource stock comes to be smaller than the one that maximizes aggregate payoff. The combination of appropriation externalities and lack of individualized and sufficiently well-defined property rights provides a classic case for government intervention, but socially optimal resource management may also be achieved by means of cooperation among resource-users.\u003c/p\u003e \u003c/div\u003e \u003cdiv id=\"Sec3\" class=\"Section2\"\u003e \u003ch2\u003e1.2. Common-pool resources\u003c/h2\u003e \u003cp\u003eThe commons or common-pool resources are resources that are difficult to exclude (i.e., it is difficult to limit their appropriation) and have high levels of subtract ability (i.e., their use or appropriation by a person or a group of people reduces the ability of others to benefit from them). When resource extraction is rival tragedy of the common follows. In essence, the tragedy of the commons arises because of an externality. Environmental degradation is a modern tragedy of the commons. When one or more individuals appropriate the common resource, it reduces the quality or quantity of the resource available for others. Examples are irrigation systems, fishers, ground water basins, grazing lands, and some forest resources. For open access common pool resources (CPR), it is hard for such resources to allocate equitably nor optimally. The remedy for this particular externality must instead be determined by the individuals affected by it, through negotiation. Users of such resources may in many instances overcome incentives to destroy the resources by developing rules that enable them utilize these resources effectively [\u003cspan citationid=\"CR2\" class=\"CitationRef\"\u003e2\u003c/span\u003e]. Potentially, there are other ways of improving the use of the resources. It would be feasible for the users that share a resource to merge.\u003c/p\u003e \u003cp\u003eCooperation can help in managing local (intra-national) resource problems. That is, as local communities often are homogeneous (composed of people who live near each other and who have intimate connections, shared values, and common histories) it is much easier to cooperate.\u003c/p\u003e \u003cp\u003eGovernment may assign resource rights to the community group, and thereby prevent entry by outsiders. Of course, free riding may make effective management of common property resources harder.\u003c/p\u003e \u003cp\u003eSome recent studies focusing on individual natural resource use behavior have tried to adopt game theory. Most studies for example, [\u003cspan citationid=\"CR3\" class=\"CitationRef\"\u003e3\u003c/span\u003e, \u003cspan citationid=\"CR4\" class=\"CitationRef\"\u003e4\u003c/span\u003e, \u003cspan citationid=\"CR5\" class=\"CitationRef\"\u003e5\u003c/span\u003e, \u003cspan citationid=\"CR6\" class=\"CitationRef\"\u003e6\u003c/span\u003e, \u003cspan citationid=\"CR7\" class=\"CitationRef\"\u003e7\u003c/span\u003e, \u003cspan citationid=\"CR8\" class=\"CitationRef\"\u003e8\u003c/span\u003e, \u003cspan citationid=\"CR9\" class=\"CitationRef\"\u003e9\u003c/span\u003e] have focused on public goods (PG) games, and analyzed the extent to which especially punishments and rewards affect behavior.\u003c/p\u003e \u003cp\u003eThis study focuses on environmental problems, especially, those related to (renewable) natural resource use, in which individual\u0026rsquo;s harvesting activity negatively affects the returns to all other users of the resource under consideration. A game that better captures this underlying mechanism is the common pool resource (CPR) game. While in some cases users may construct formal or in formal institutions to more efficiently manage the commons. Generally such unexclusive resources are susceptible for over-consumption which referred as, the \u0026ldquo;tragedy of the commons\u0026rdquo;.\u003c/p\u003e \u003cp\u003eThis paper briefly describes the basis for common pool resource game models compatible with the real-world environmental problems; show mathematically how the individual level of extraction effort differs from the socially optimal. Provide a cooperation framework in which individual\u0026rsquo;s choose the social welfare maximizing level of extraction effort over the unilateral payoff maximizing level. And argue the simplest framework used for the economic analysis of CPR.\u003c/p\u003e \u003c/div\u003e"},{"header":"2. Set Up Of The Common Pool Resource Game","content":"\u003cp\u003eThe basis for an open access natural resource extraction oriented game is the finitely repeated common pool resource game similar to that of [10⦌. Let there is a closed community of \u003cem\u003eN\u003c/em\u003e resource users, where \u003cem\u003eN\u0026thinsp;\u0026gt;\u0026thinsp;1\u003c/em\u003e and the community members have unrestricted access to the common pool resource. In every period \u003cem\u003et\u0026thinsp;=\u0026thinsp;1, \u0026hellip;, T\u003c/em\u003e, each user \u003cem\u003ei\u0026thinsp;=\u0026thinsp;1,. . ., N\u003c/em\u003e can allocate a fixed endowment of effort, \u003cem\u003ee\u0026thinsp;\u0026gt;\u0026thinsp;0\u003c/em\u003e, between CPR extraction and an alternative economic activity (the outside option). Assuming extraction of CPR by individuals who have access to the open access resource increases in effort level, where extraction effort exerted by user \u003cem\u003ei\u003c/em\u003e in period \u003cem\u003et\u003c/em\u003e is denoted by \u003cem\u003ex\u003c/em\u003e\u003csub\u003e\u003cem\u003ei,t\u003c/em\u003e\u003c/sub\u003e, and hence user \u003cem\u003ei\u003c/em\u003e\u0026rsquo;s effort devoted to the outside option equals (\u003cem\u003ee-x\u003c/em\u003e\u003csub\u003e\u003cem\u003ei,t\u003c/em\u003e\u003c/sub\u003e). The outside option yields a fixed per-unit wage rate, w. When exerting extraction effort, users incur cost that is linear in extraction effort. The marginal cost is assumed to be constant and equal to c. In addition to the above assumptions, this paper assumes complete information throughout.\u003c/p\u003e \u003cp\u003eAssume again the group\u0026rsquo;s benefit in period \u003cem\u003et\u003c/em\u003e, \u003cem\u003eB\u003c/em\u003e\u003csub\u003e\u003cem\u003et\u003c/em\u003e\u003c/sub\u003e, depends on the aggregate amount of extraction effort in that period, \u003cspan class=\"InlineEquation\"\u003e\u003cspan class=\"mathinline\"\u003e\\({X}_{t }=\\sum _{i=1}^{N}{x}_{i,t}\\)\u003c/span\u003e\u003c/span\u003e, where price is assumed constant and equal to 1. Let the benefit function is given by \u003cspan class=\"InlineEquation\"\u003e\u003cspan class=\"mathinline\"\u003e\\({B}_{t}\\left({X}_{t}\\right)\\)\u003c/span\u003e\u003c/span\u003e=\u003cspan class=\"InlineEquation\"\u003e\u003cspan class=\"mathinline\"\u003e\\({ aX}_{t}\\)\u003c/span\u003e\u003c/span\u003e-\u003cspan class=\"InlineEquation\"\u003e\u003cspan class=\"mathinline\"\u003e\\({bX}_{t}^{2}\\)\u003c/span\u003e\u003c/span\u003e, for any positive constants a and b. User \u003cem\u003ei\u003c/em\u003e\u0026rsquo;s share in these benefits is proportional to his/her share in aggregate extraction effort (\u003cspan class=\"InlineEquation\"\u003e\u003cspan class=\"mathinline\"\u003e\\(\\frac{{x}_{i,t}}{{X}_{t}}\\)\u003c/span\u003e\u003c/span\u003e). Hence, user \u003cem\u003ei\u003c/em\u003e\u0026rsquo;s aggregate payoff, \u0026#120561;\u003csub\u003ei,t\u003c/sub\u003e in period \u003cem\u003et\u003c/em\u003e equals:\u003cdiv id=\"Equa\" class=\"Equation\"\u003e\u003cdiv format=\"TEX\" class=\"mathdisplay\" id=\"FileID_Equa\" name=\"EquationSource\"\u003e\n$${\\varPi }_{i,t}\\left({x}_{i,t},{X}_{t}\\right)=w\\left(e-{x}_{i,t}\\right)+\\frac{{x}_{i,t}}{{X}_{t}} \\left({aX}_{t}-{bX}_{t}^{2}\\right)-{cx}_{i,t} \\left(1\\right)$$\u003c/div\u003e\u003c/div\u003e\u003c/p\u003e \u003cp\u003eFor the sake of simplicity, let\u0026rsquo;s limit the common pool resource extraction game to the one shot game. And the socially optimal extraction effort is the one that maximizes the unweighted payoffs of all \u003cem\u003eN\u003c/em\u003e users in the group defined in Eq.\u0026nbsp;(1). In which the solution concept refers to the sort of cooperative games that reflect actions taken in a coordinated way by all parties.\u003c/p\u003e \u003cp\u003e \u003cstrong\u003eProposition 1\u003c/strong\u003e \u003cp\u003eFor unregulated open access common pool resource game, every user \u003cem\u003ei\u003c/em\u003e in the community chooses an equilibrium level of harvesting effort that exceeds the social optimum.\u003c/p\u003e \u003c/p\u003e \u003cp\u003e \u003cstrong\u003eProof\u003c/strong\u003e \u003cp\u003eIn order to show whether the equitable socially optimal extraction effort level is below the individual unilateral equilibrium level of effort, first it is important to specify the net benefit (payoff) of all \u003cem\u003eN -\u003c/em\u003e users in the group:\u003cdiv id=\"Equb\" class=\"Equation\"\u003e\u003cdiv format=\"TEX\" class=\"mathdisplay\" id=\"FileID_Equb\" name=\"EquationSource\"\u003e\n$$\\varPi \\left({X}_{t}\\right)=w\\left(Ne-{X}_{t}\\right)+{aX}_{t}-{bX}_{t}^{2} -{cX}_{t} \\left(2\\right)$$\u003c/div\u003e\u003c/div\u003e\u003c/p\u003e \u003c/p\u003e \u003cp\u003eWe can apply the first order derivative on Eq.\u0026nbsp;(2) to obtain the aggregate optimal extraction level of effort.\u003cdiv id=\"Equc\" class=\"Equation\"\u003e\u003cdiv format=\"TEX\" class=\"mathdisplay\" id=\"FileID_Equc\" name=\"EquationSource\"\u003e\n$$\\frac{\\partial \\varPi \\left({X}_{t}\\right) }{\\partial {X}_{t}}=\\frac{\\partial \\left[\\left({Ne-X}_{t}\\right)w+{aX}_{t}-{bX}_{t}^{2} -{cX}_{t}\\right]}{\\partial {X}_{t}}=0$$\u003c/div\u003e\u003c/div\u003e\u003c/p\u003e \u003cp\u003eHence, the optimal extraction effort level,\u003cspan class=\"InlineEquation\"\u003e\u003cspan class=\"mathinline\"\u003e\\({X}_{t}=\\)\u003c/span\u003e\u003c/span\u003e \u003cspan class=\"InlineEquation\"\u003e\u003cspan class=\"mathinline\"\u003e\\({X}^{*}=\\frac{a-c-w }{2b}\\)\u003c/span\u003e\u003c/span\u003e, for \u003cspan class=\"InlineEquation\"\u003e\u003cspan class=\"mathinline\"\u003e\\(a-c-w\u0026gt;0\\)\u003c/span\u003e\u003c/span\u003e. And the individual equitable socially optimal extraction effort level is:\u003c/p\u003e \u003cp\u003e \u003cspan class=\"InlineEquation\"\u003e \u003cspan class=\"mathinline\"\u003e\\({{x}_{i,t}=x}^{*}=\\frac{{X}^{*}}{N}=\\frac{a-c-w }{2Nb}\\)\u003c/span\u003e \u003c/span\u003e(a)\u003c/p\u003e \u003cp\u003eLet us use \u003cspan class=\"InlineEquation\"\u003e\u003cspan class=\"mathinline\"\u003e\\({x}_{0}\\)\u003c/span\u003e\u003c/span\u003e to denote common pool resource game in the absence of peer regulation. Assuming that individuals are rational and aim to maximize their own payoffs, user \u003cem\u003ei\u0026rsquo;s\u003c/em\u003e best response function to any level of aggregate extraction effort by all others (\u003cspan class=\"InlineEquation\"\u003e\u003cspan class=\"mathinline\"\u003e\\({X}_{-i,t}=\\sum _{j\\ne i}^{N}{x}_{j,t}\\)\u003c/span\u003e\u003c/span\u003e) is obtained by rearranging Eq.\u0026nbsp;(1) and apply first order condition (foc).\u003cdiv id=\"Equd\" class=\"Equation\"\u003e\u003cdiv format=\"TEX\" class=\"mathdisplay\" id=\"FileID_Equd\" name=\"EquationSource\"\u003e\n$${\\varPi }_{i,t}\\left({x}_{i,t},{X}_{t}\\right)=w\\left(e-{x}_{i,t}\\right)+\\frac{{x}_{i,t}}{{X}_{t}} \\left({aX}_{t}-{bX}_{t}^{2}\\right)-{cx}_{i,t}$$\u003c/div\u003e\u003c/div\u003e\u003c/p\u003e \u003cp\u003e \u003cspan class=\"InlineEquation\"\u003e \u003cspan class=\"mathinline\"\u003e\\(= w\\left(e-{x}_{i,t}\\right)+{x}_{i,t} \\left(a-b{X}_{t}\\right)-c{x}_{i,t}\\)\u003c/span\u003e \u003c/span\u003e, substituting \u003cspan class=\"InlineEquation\"\u003e\u003cspan class=\"mathinline\"\u003e\\({X}_{t}={x}_{i,t}+{X}_{-i,t}\\)\u003c/span\u003e\u003c/span\u003e\u003cdiv id=\"Eque\" class=\"Equation\"\u003e\u003cdiv format=\"TEX\" class=\"mathdisplay\" id=\"FileID_Eque\" name=\"EquationSource\"\u003e\n$${\\varPi }_{i,t}\\left({x}_{i,t},{X}_{-i,t}\\right)=w\\left(e-{x}_{i,t}\\right)+{ax}_{i,t}-{bx}_{i,t}^{2} -b{x}_{i,t} {X}_{-i,t}-c{x}_{i,t}$$\u003c/div\u003e\u003c/div\u003e\u003c/p\u003e \u003cp\u003eLet\u0026rsquo;s apply foc with respect to \u003cspan class=\"InlineEquation\"\u003e\u003cspan class=\"mathinline\"\u003e\\({x}_{i,t}\\)\u003c/span\u003e\u003c/span\u003e on this version of Eq.\u0026nbsp;(1):\u003c/p\u003e \u003cp\u003e \u003cspan class=\"InlineEquation\"\u003e \u003cspan class=\"mathinline\"\u003e\\(\\frac{\\partial {\\varPi }_{i,t}\\left({x}_{i,t}, {X}_{-i,t}\\right) }{\\partial {x}_{i,t}}=\\frac{\\partial w\\left(e -{ x}_{i,t}\\right)+{ ax}_{i,t}-{ bx}_{i,t}^{2} - b{x}_{i,t} {X}_{-i,t}- c{x}_{i,t}}{\\partial {x}_{i,t} }\\)\u003c/span\u003e \u003c/span\u003e= 0\u003c/p\u003e \u003cp\u003e \u003cspan class=\"InlineEquation\"\u003e \u003cspan class=\"mathinline\"\u003e\\(=a-w-2b{x}_{i,t}-b{X}_{-i,t}-c=0\\)\u003c/span\u003e \u003c/span\u003e, and\u003c/p\u003e \u003cp\u003e \u003cspan class=\"InlineEquation\"\u003e \u003cspan class=\"mathinline\"\u003e\\({ x}_{i,t}\\left({X}_{-i,t}\\right)=\\frac{a - c - w}{2b}-\\frac{{X}_{-i,t}}{2}\\)\u003c/span\u003e \u003c/span\u003e (b)\u003c/p\u003e \u003cp\u003eFor a symmetric individual extraction effort,\u003cspan class=\"InlineEquation\"\u003e\u003cspan class=\"mathinline\"\u003e\\({ x}_{i}\\)\u003c/span\u003e\u003c/span\u003e we have \u003cspan class=\"InlineEquation\"\u003e\u003cspan class=\"mathinline\"\u003e\\({ x}_{i}={x}_{j}=x\\)\u003c/span\u003e\u003c/span\u003e and \u003cspan class=\"InlineEquation\"\u003e\u003cspan class=\"mathinline\"\u003e\\({X}_{-i,t}=\\sum _{j\\ne i}^{N}{x}_{j,t }=(\\text{N}-1){x}_{i,t}\\)\u003c/span\u003e\u003c/span\u003e. Substituting \u003cspan class=\"InlineEquation\"\u003e\u003cspan class=\"mathinline\"\u003e\\({X}_{-i,t}=(\\text{N}-1){x}_{i,t}\\)\u003c/span\u003e\u003c/span\u003e in the first order condition equation (b) above we obtain:\u003cdiv id=\"Equf\" class=\"Equation\"\u003e\u003cdiv format=\"TEX\" class=\"mathdisplay\" id=\"FileID_Equf\" name=\"EquationSource\"\u003e\n$${ x}_{i,t}=\\frac{a - c - w}{2b}-\\frac{{\\left(N- 1\\right)x}_{i,t}}{2}$$\u003c/div\u003e\u003c/div\u003e\u003cdiv id=\"Equg\" class=\"Equation\"\u003e\u003cdiv format=\"TEX\" class=\"mathdisplay\" id=\"FileID_Equg\" name=\"EquationSource\"\u003e\n$${ x}_{i,t} +\\frac{{\\left(N-1\\right)x}_{i,t}}{2}=\\frac{a - c - w}{2b}$$\u003c/div\u003e\u003c/div\u003e\u003cdiv id=\"Equh\" class=\"Equation\"\u003e\u003cdiv format=\"TEX\" class=\"mathdisplay\" id=\"FileID_Equh\" name=\"EquationSource\"\u003e\n$${ x}_{i,t}\\left(1+\\frac{N-1}{2}\\right)=\\frac{a - c - w}{2b}$$\u003c/div\u003e\u003c/div\u003e\u003c/p\u003e \u003cp\u003eThus\u003cspan class=\"InlineEquation\"\u003e\u003cspan class=\"mathinline\"\u003e\\({ x}_{i,t}={x}_{j,t}={x}_{0}^{NE}=\\)\u003c/span\u003e\u003c/span\u003e \u003cspan class=\"InlineEquation\"\u003e\u003cspan class=\"mathinline\"\u003e\\(\\frac{a - c - w}{b(N+1)}\\)\u003c/span\u003e\u003c/span\u003e (c)\u003c/p\u003e \u003cp\u003eFrom equations (a) and (b) above compare the socially optimal level of extraction effort (\u003cspan class=\"InlineEquation\"\u003e\u003cspan class=\"mathinline\"\u003e\\({x}^{*})\\)\u003c/span\u003e\u003c/span\u003e for individuals in the group with the unique symmetric Nash equilibrium extraction effort\u003cspan class=\"InlineEquation\"\u003e\u003cspan class=\"mathinline\"\u003e\\({x}_{0}^{NE}\\)\u003c/span\u003e\u003c/span\u003e, \u003cspan class=\"InlineEquation\"\u003e\u003cspan class=\"mathinline\"\u003e\\({x}_{0}^{NE}\\)\u003c/span\u003e\u003c/span\u003e\u0026gt;\u003cspan class=\"InlineEquation\"\u003e\u003cspan class=\"mathinline\"\u003e\\({ x}^{*}\\)\u003c/span\u003e\u003c/span\u003e for all \u003cem\u003eN \u0026gt;\u0026thinsp;1\u003c/em\u003e, and \u003cspan class=\"InlineEquation\"\u003e\u003cspan class=\"mathinline\"\u003e\\(a-c-w\u0026gt;0\\)\u003c/span\u003e\u003c/span\u003e. This implies that the unregulated common pool resource game poses a social dilemma. Social dilemma indicates for the situation in which individuals\u0026rsquo; and society\u0026rsquo;s interest is in conflict.\u003c/p\u003e \u003cp\u003eIn the real world, for open access resources for e.g., forest, fishery, fresh water, grazing land and mining when the number of users for the unrestricted resource increased, every additional extraction by new users increases the amount of time and effort required to extract extra resource. In Fig.\u0026nbsp;\u003cspan refid=\"Fig1\" class=\"InternalRef\"\u003e1\u003c/span\u003e under, social benefits and costs of forest extraction (e.g., fuel wood, charcoal or grass) for the benefit function specified above are illustrated. Total benefits of extraction are calculated by multiplying price per unit by the amount of harvest. The marginal benefit curve is down ward sloping because the greater the amount of extraction effort expended, the smaller the resulting remaining resource will be. And the smaller left over resource, the smaller harvest per unit of effort expended.\u003c/p\u003e \u003cp\u003eThe optimal level of extraction effort level, \u003cspan class=\"InlineEquation\"\u003e\u003cspan class=\"mathinline\"\u003e\\({x}^{*}\\)\u003c/span\u003e\u003c/span\u003e in Fig.\u0026nbsp;\u003cspan refid=\"Fig1\" class=\"InternalRef\"\u003e1\u003c/span\u003e (b) is the level where, the marginal benefit would be just equal to the marginal social cost, implying that net benefits are maximized. The marginal social cost includes not only the private operating costs but also the social cost of depleting the open access resource. With all extractors having completely unrestricted access to the resource the resulting allocation would be above the optimal level. As Fig.\u0026nbsp;\u003cspan refid=\"Fig1\" class=\"InternalRef\"\u003e1\u003c/span\u003e (a) shows, individuals would exploit the resource until their total benefit equaled total cost; or equivalently as Fig.\u0026nbsp;\u003cspan refid=\"Fig1\" class=\"InternalRef\"\u003e1\u003c/span\u003e (b) shows until their marginal benefit equals private marginal cost, implied by the level of effort\u003cspan class=\"InlineEquation\"\u003e\u003cspan class=\"mathinline\"\u003e\\({ x}_{0}\\)\u003c/span\u003e\u003c/span\u003e. Excessive exploitation of the resource occurs because individuals have no incentive to leave resource after.\u003c/p\u003e \u003cp\u003eIn the presence of sufficient demand for the common pool resource, unrestricted access will cause resource to be over exploited. This is because the individuals/groups exploiting the open access would not have any incentive to conserve as the benefits derived from restraint would, to some extent, be captured by other extractors.\u003c/p\u003e \u003cp\u003eHence, it is essential for the group who have unlimited resource access to look for a mechanism that help in addressing the social dilemma in common pool resources. The dilemma is characterized by the sub-optimal equilibrium of non-cooperative game associated with an individual\u0026rsquo;s decision that only satisfies their own selfish interest. Thus, devising an arrangement that may enforce users select the socially optimal level of effort instead of the business as usual or (unregulated) effort level is suggested in our framework. That is, the arrangement is required to encompass a kind of agreement that motivate individuals to self-select the optimal (cooperative) level of extraction effort over the business as usual effort level.\u003c/p\u003e \u003cp\u003eFor a static common-pool resource, every player (user) extracts too much and leaves smaller pie for the other users. If we extend the static CPR extraction game in to a simple two stage dynamic common-pool problem, efficiency may further reduced. Every user may be incentivized to extract a lot at the first stage and this will discourage other from extracting at the second stage. The outcome is that even more of the resource may be extracted at the beginning. Hence, unless the users introduced some regulatory framework and act accordingly, intertemporal allocation of unregulated common pool resource becomes more inefficient: too much is extracted in the beginning and too little later on (conditional on the total amount extracted in the beginning).\u003c/p\u003e"},{"header":"3. The Infinitely Repeated Common Pool Resource Extraction Game","content":"\u003cp\u003eFor static common pool resource game users make extraction effort choice once simultaneously. In reality as individual user live together and reap resource benefit for longer period of time, strategic decisions are required to take this in to account. In contrast to the situation where agents interact only once, any mutually beneficial outcome can be sustained as equilibrium when agents interact repeatedly and frequently [11⦌. Formally, repeated games refer to a class of models where the same set of agents repeatedly plays the same game, called the \u0026lsquo;stage game\u0026rsquo;, over a long time horizon. Behavioral studies indicate that the equilibrium strategy for a finitely repeated games or games with certain number of rounds is the same as the equilibrium of the one shot game. In the absence of binding agreements, infinitely repeated interactions allow for socially beneficial outcomes. That is, if interactions are infinitely repeated, then socially beneficial outcomes that cannot be sustained by players with short-term objectives may sustained by players with long-term objectives.\u003c/p\u003e \u003cp\u003eAlthough one may argue that players do not really live infinitely long (so that the finite horizon case is more realistic), there is a good reason to consider the infinite horizon models. Even though the time horizon is finite, if players do not know in advance exactly when the game ends, the situation can be formulated as an infinitely repeated game.\u003c/p\u003e \u003cp\u003eIn a general game, any outcome which Pareto dominates the Nash equilibrium can be sustained by a strategy which reverts to the Nash equilibrium after a deviation. Such a strategy is called a trigger strategy. And the strategies with payoff sets Pareto dominate the Nash equilibrium may ascertain by the Folk theorem. The Folk theorem asserts that when players are equally and sufficiently patient, all payoffs that are socially feasible and Pareto dominate the minimax are subgame perfect equilibrium payoffs. Alternatively the Folk theorem may be defined as: for some discount factor sufficiently close to unity, any set of feasible payoffs that are preferred by all individuals to the Nash equilibrium payoffs can be supported as subgame perfect equilibria for the repeated game.\u003c/p\u003e \u003cp\u003eHence, for infinitely repeated game, cooperation may sustain through a self-enforcing agreements provided that individuals are sufficiently patient. This is in line with Barrett [\u003cspan citationid=\"CR12\" class=\"CitationRef\"\u003e12\u003c/span\u003e] who suggested the possibility of enforcing cooperation for international environmental problems. The repeated game differs from its one-shot version that individuals who deviate from a promise to play cooperation can be punished by members of the group. Suppose that the game is played forever and discount rates are vanishingly small. Suppose, moreover, that the players play the so-called \u0026ldquo;Triger\u0026rdquo; strategy- each begins by playing low effort. Thereafter, each player plays low effort provided no player played high effort in the past. Otherwise, each individual in the group apply high effort.\u003c/p\u003e \u003cdiv id=\"Sec6\" class=\"Section2\"\u003e \u003ch2\u003e3.1. Infinitely repeated common pool resource extraction stage game\u003c/h2\u003e \u003cp\u003eLet the common pool resource game is a repeated game consists of a stage game and a set of times when the stage game is played. Assume, there are N- players (users) indexed by \u003cspan class=\"InlineEquation\"\u003e\u003cspan class=\"mathinline\"\u003e\\(i or j\\in N=\\left\\{1,\\dots ,n\\right\\}.\\)\u003c/span\u003e\u003c/span\u003e Assume again each individual writes agreement with at most one group of players, the coalition to which he/she belongs at stage 1. For the sake of simplicity, we assume only two possible extraction efforts\u003cspan class=\"InlineEquation\"\u003e\u003cspan class=\"mathinline\"\u003e\\({x}_{i}\\in \\{ \\underset{\\_}{x} ,\\stackrel{-}{x}\\}\\)\u003c/span\u003e\u003c/span\u003e. Where, \u003cspan class=\"InlineEquation\"\u003e\u003cspan class=\"mathinline\"\u003e\\(\\underset{\\_}{x}\\)\u003c/span\u003e\u003c/span\u003eand \u003cspan class=\"InlineEquation\"\u003e\u003cspan class=\"mathinline\"\u003e\\(\\stackrel{-}{x}\\)\u003c/span\u003e\u003c/span\u003e stands for relatively low and high extraction efforts respectively. Assume again at the harvesting stage (stage 2) users who have access to the resource simultaneously decide between more or less effort, \u003cspan class=\"InlineEquation\"\u003e\u003cspan class=\"mathinline\"\u003e\\({x}_{i}\\in \\{ \\underset{\\_}{x} ,\\stackrel{-}{x}\\}\\)\u003c/span\u003e\u003c/span\u003e.\u003c/p\u003e \u003cp\u003eLet in the unregulated harvesting effort individuals decision constitute a prisoners\u0026rsquo; dilemma game. That is, individuals are benefited from more own harvesting effort for any fixed harvesting effort level from all other players, \u003cspan class=\"InlineEquation\"\u003e\u003cspan class=\"mathinline\"\u003e\\({X}_{-i}=\\sum _{j\\ne i}^{N}{x}_{j,}\\)\u003c/span\u003e\u003c/span\u003e but every individual would be better off if everyone else have less instead of more effort. The prisoners\u0026rsquo; dilemma game is a reasonable stage game in this game of extraction effort for common pool resources. If \u003cspan class=\"InlineEquation\"\u003e\u003cspan class=\"mathinline\"\u003e\\({B(x}_{i})\\)\u003c/span\u003e\u003c/span\u003e and \u003cspan class=\"InlineEquation\"\u003e\u003cspan class=\"mathinline\"\u003e\\(c\\)\u003c/span\u003e\u003c/span\u003e are the benefit of extraction effort and marginal cost of extraction efforts respectively, net payoff from extraction is given by:\u003cdiv id=\"Equ1\" class=\"Equation\"\u003e\u003cdiv format=\"TEX\" class=\"mathdisplay\" id=\"FileID_Equ1\" name=\"EquationSource\"\u003e\n$${\\varPi }_{i}\\left({x}_{i}\\right)=B\\left({x}_{i}\\right)-c\\sum _{i=1}^{N}{x}_{i}$$\u003c/div\u003e\u003cdiv class=\"EquationNumber\"\u003e3\u003c/div\u003e\u003c/div\u003e\u003c/p\u003e \u003cp\u003eHence for \u003cspan class=\"InlineEquation\"\u003e\u003cspan class=\"mathinline\"\u003e\\({x}_{i}\\in \\{ \\underset{\\_}{x} ,\\stackrel{-}{x}\\}\\)\u003c/span\u003e\u003c/span\u003e, less extraction effort, \u003cspan class=\"InlineEquation\"\u003e\u003cspan class=\"mathinline\"\u003e\\(x=\\underset{\\_}{x}\\)\u003c/span\u003e\u003c/span\u003e will simply be the individually optimal extraction effort if the one period net payoff from less extraction effort outweighs that of more extraction effort.\u003c/p\u003e \u003cp\u003e \u003cspan class=\"InlineEquation\"\u003e \u003cspan class=\"mathinline\"\u003e\\(B\\left({\\stackrel{-}{x}}_{i}\\right)-c\\sum _{i=1}^{N}{\\stackrel{-}{x}}_{i}\\)\u003c/span\u003e \u003c/span\u003e \u0026lt; \u003cspan class=\"InlineEquation\"\u003e\u003cspan class=\"mathinline\"\u003e\\(B\\left({\\underset{\\_}{x}}_{i}\\right)-c\\sum _{i=1}^{N}{\\underset{\\_}{x}}_{i}\\)\u003c/span\u003e\u003c/span\u003e\u003c/p\u003e \u003cp\u003eAssuming all individuals in the group exert same extraction effort at optimality, the above equation may be written as:\u003c/p\u003e \u003cp\u003e \u003cspan class=\"InlineEquation\"\u003e \u003cspan class=\"mathinline\"\u003e\\(B\\left({\\stackrel{-}{x}}_{i}\\right)-cN\\stackrel{-}{x}\\)\u003c/span\u003e \u003c/span\u003e \u0026lt; \u003cspan class=\"InlineEquation\"\u003e\u003cspan class=\"mathinline\"\u003e\\(B\\left({\\underset{\\_}{x}}_{i}\\right)-cN\\underset{\\_}{x}\\)\u003c/span\u003e\u003c/span\u003e\u003cdiv id=\"Equi\" class=\"Equation\"\u003e\u003cdiv format=\"TEX\" class=\"mathdisplay\" id=\"FileID_Equi\" name=\"EquationSource\"\u003e\n$$B\\left({\\stackrel{-}{x}}_{i}\\right)- B\\left({\\underset{\\_}{x}}_{i}\\right)\u0026lt;cN(\\stackrel{-}{x}-\\underset{\\_}{x})$$\u003c/div\u003e\u003c/div\u003e\u003c/p\u003e \u003cp\u003e \u003cspan class=\"InlineEquation\"\u003e \u003cspan class=\"mathinline\"\u003e\\(\\frac{B\\left({\\stackrel{-}{x}}_{i}\\right)- B\\left({ \\underset{\\_}{x}}_{i}\\right)}{c(\\stackrel{-}{x}-\\underset{\\_}{x}) } \u0026lt;N\\)\u003c/span\u003e \u003c/span\u003e (d)\u003c/p\u003e \u003cp\u003eSimilarly, high extraction effort \u003cspan class=\"InlineEquation\"\u003e\u003cspan class=\"mathinline\"\u003e\\(x=\\stackrel{-}{x}\\)\u003c/span\u003e\u003c/span\u003e will be a dominant strategy if the net payoff from more extraction effort exceeds that of less extraction effort, taking in to account only individuals own extraction cost.\u003c/p\u003e \u003cp\u003e \u003cspan class=\"InlineEquation\"\u003e \u003cspan class=\"mathinline\"\u003e\\(B\\left({\\stackrel{-}{x}}_{i}\\right)-c{\\overline{x}}_{i}\\)\u003c/span\u003e \u003c/span\u003e \u0026gt; \u003cspan class=\"InlineEquation\"\u003e\u003cspan class=\"mathinline\"\u003e\\(B\\left({\\underset{\\_}{x}}_{i}\\right)-\\)\u003c/span\u003e\u003c/span\u003ec\u003cspan class=\"InlineEquation\"\u003e\u003cspan class=\"mathinline\"\u003e\\({\\underset{\\_}{x}}_{i}\\)\u003c/span\u003e\u003c/span\u003e\u003c/p\u003e \u003cp\u003e \u003cspan class=\"InlineEquation\"\u003e \u003cspan class=\"mathinline\"\u003e\\(B\\left({\\stackrel{-}{x}}_{i}\\right)-\\)\u003c/span\u003e \u003c/span\u003e \u003cspan class=\"InlineEquation\"\u003e\u003cspan class=\"mathinline\"\u003e\\(B\\left({\\underset{\\_}{x}}_{i}\\right)\\)\u003c/span\u003e\u003c/span\u003e \u0026gt;\u003cspan class=\"InlineEquation\"\u003e\u003cspan class=\"mathinline\"\u003e\\(c{\\overline{x}}_{i}-\\)\u003c/span\u003e\u003c/span\u003e c\u003cspan class=\"InlineEquation\"\u003e\u003cspan class=\"mathinline\"\u003e\\({\\underset{\\_}{x}}_{i}\\)\u003c/span\u003e\u003c/span\u003e\u003c/p\u003e \u003cp\u003e \u003cspan class=\"InlineEquation\"\u003e \u003cspan class=\"mathinline\"\u003e\\(\\frac{B\\left({\\stackrel{-}{x}}_{i}\\right)- B\\left({\\underset{\\_}{x}}_{i}\\right)}{c({\\stackrel{-}{x}}_{i}- {\\underset{\\_}{x}}_{i})}\u0026gt;1\\)\u003c/span\u003e \u003c/span\u003e (e)\u003c/p\u003e \u003cp\u003eThus, the common pool resource extraction effort game is a prisoner\u0026rsquo;s dilemma game if both (d) and (e) holds true. And from equations (d) and (e) we reach at the expression under.\u003cdiv id=\"Equ2\" class=\"Equation\"\u003e\u003cdiv format=\"TEX\" class=\"mathdisplay\" id=\"FileID_Equ2\" name=\"EquationSource\"\u003e\n$$1\u0026lt;\\frac{B\\left({\\stackrel{-}{x}}_{i}\\right)- B\\left({\\underset{\\_}{x}}_{i}\\right)}{c({\\stackrel{-}{x}}_{i}- {\\underset{\\_}{x}}_{i})}\u0026lt;N$$\u003c/div\u003e\u003cdiv class=\"EquationNumber\"\u003e4\u003c/div\u003e\u003c/div\u003e\u003c/p\u003e \u003cp\u003eIf the users agree to cooperate they can reach a better equilibrium than the business as usual equilibrium that happens in the Prisoner\u0026rsquo;s dilemma game. In order for cooperation \u003cspan class=\"InlineEquation\"\u003e\u003cspan class=\"mathinline\"\u003e\\(x=\\underset{\\_}{x}\\)\u003c/span\u003e\u003c/span\u003e sustain as a Pareto optimal subgame perfect equilibrium, the worst equilibrium, \u003cspan class=\"InlineEquation\"\u003e\u003cspan class=\"mathinline\"\u003e\\(x=\\stackrel{-}{x}\\)\u003c/span\u003e\u003c/span\u003e might be used as threat to enforce better equilibrium. That is, \u003cspan class=\"InlineEquation\"\u003e\u003cspan class=\"mathinline\"\u003e\\(x=\\stackrel{-}{x}\\)\u003c/span\u003e\u003c/span\u003e can be sustain as a trigger strategy in a subgame perfect equilibrium (SPE), where any deviation requires the players to revert to the business as usual equilibrium forever. This is in line to Barrett [12⦌ where he suggested a strategy of reciprocity to enforce compliance for international environmental agreements.\u003c/p\u003e \u003cp\u003e \u003cstrong\u003eProposition 2\u003c/strong\u003e\u003c/p\u003e \u003cp\u003eCooperation hardly sustains if:\u003c/p\u003e \u003c/p\u003e \u003cp\u003ei. Individuals are heavily discounting the value of cooperating in the future (i.e., \u0026#120575; is small).\u003c/p\u003e \u003cp\u003eii. the number of individuals in the group is relatively few\u003c/p\u003e \u003cp\u003eProof for proposition \u003cspan refid=\"FPar3\" class=\"InternalRef\"\u003e2\u003c/span\u003e (i):\u003c/p\u003e \u003cp\u003eFor an infinitely repeated game with discount rate, r and discount factor, \u003cspan class=\"InlineEquation\"\u003e\u003cspan class=\"mathinline\"\u003e\\(\\delta =\\frac{1}{1+r}\\)\u003c/span\u003e\u003c/span\u003e, in the absence of binding agreements either due to the absence of third party to enforce agreements or because enforcing agreement by a third party is costly, self-enforcing agreement may sustain if the discounted payoff streams from cooperation (\u003cspan class=\"InlineEquation\"\u003e\u003cspan class=\"mathinline\"\u003e\\({V}_{i}^{c}\\)\u003c/span\u003e\u003c/span\u003e) out weight the payoff streams from breaking cooperation (\u003cspan class=\"InlineEquation\"\u003e\u003cspan class=\"mathinline\"\u003e\\({V}_{i}^{d}\\)\u003c/span\u003e\u003c/span\u003e). Where, the discounted payoff stream for individual i is, \u003cspan class=\"InlineEquation\"\u003e\u003cspan class=\"mathinline\"\u003e\\({V}_{i}=\\sum _{t=0}^{\\infty }{\\delta }^{t-1}{\\varPi }_{it}\\)\u003c/span\u003e\u003c/span\u003e, for 0 \u0026lt; \u0026#120575; \u0026lt; 1\u003c/p\u003e \u003cp\u003e \u003cspan class=\"InlineEquation\"\u003e \u003cspan class=\"mathinline\"\u003e\\({V}_{i}^{c}=\\frac{B\\left(\\underset{\\_}{x}\\right) - cN\\underset{\\_}{x}}{(1 - \\delta )}= ; {V}_{i}^{d}=\\text{B}\\left(\\stackrel{-}{x}\\right)-c\\stackrel{-}{x}-c\\left(N-1\\right)\\underset{\\_}{x}+\\frac{\\delta [\\text{B}\\left(\\stackrel{-}{x}\\right) - cN\\stackrel{-}{x})] }{(1 - \\delta )}\\)\u003c/span\u003e \u003c/span\u003e, mathematically this may be specified as:\u003c/p\u003e \u003cp\u003e \u003cspan class=\"InlineEquation\"\u003e \u003cspan class=\"mathinline\"\u003e\\({V}_{i}^{c}\\ge\\)\u003c/span\u003e \u003c/span\u003e \u003cspan class=\"InlineEquation\"\u003e\u003cspan class=\"mathinline\"\u003e\\({V}_{i}^{d}\\)\u003c/span\u003e\u003c/span\u003e\u003cdiv id=\"Equj\" class=\"Equation\"\u003e\u003cdiv format=\"TEX\" class=\"mathdisplay\" id=\"FileID_Equj\" name=\"EquationSource\"\u003e\n$$\\frac{B\\left(\\underset{\\_}{x}\\right) - cN\\underset{\\_}{x}}{(1 - \\delta )}\\ge \\text{B}\\left(\\stackrel{-}{x}\\right)-c\\stackrel{-}{x}-c\\left(N-1\\right)\\underset{\\_}{x}+\\frac{\\delta [\\text{B}\\left(\\stackrel{-}{x}\\right) - cN\\stackrel{-}{x})] }{(1 - \\delta )}$$\u003c/div\u003e\u003c/div\u003e\u003cdiv id=\"Equk\" class=\"Equation\"\u003e\u003cdiv format=\"TEX\" class=\"mathdisplay\" id=\"FileID_Equk\" name=\"EquationSource\"\u003e\n$$\\frac{B\\left(\\underset{\\_}{x}\\right) - cN\\underset{\\_}{x}}{(1 - \\delta )}\\ge \\text{B}\\left(\\stackrel{-}{x}\\right)-c\\left[\\stackrel{-}{x}+\\left(N-1\\right)\\underset{\\_}{x}\\right]+\\frac{\\delta [\\text{B}\\left(\\stackrel{-}{x}\\right) - cN\\stackrel{-}{x})] }{(1 - \\delta )}$$\u003c/div\u003e\u003c/div\u003e\u003c/p\u003e \u003cp\u003e \u003cspan class=\"InlineEquation\"\u003e \u003cspan class=\"mathinline\"\u003e\\(\\text{B}\\left(\\stackrel{-}{x}\\right)-B(\\underset{\\_}{x}\\)\u003c/span\u003e \u003c/span\u003e) \u003cspan class=\"InlineEquation\"\u003e\u003cspan class=\"mathinline\"\u003e\\(\\le \\text{c}\\left(\\stackrel{-}{x} -\\underset{\\_}{x}\\right)\\left[\\delta N+\\left(1-\\delta \\right)\\right]\\)\u003c/span\u003e\u003c/span\u003e\u003c/p\u003e \u003cp\u003e \u003cspan class=\"InlineEquation\"\u003e \u003cspan class=\"mathinline\"\u003e\\({\\delta }\\ge\\)\u003c/span\u003e \u003c/span\u003e \u003cspan class=\"InlineEquation\"\u003e\u003cspan class=\"mathinline\"\u003e\\(\\widehat{\\delta }\\equiv \\frac{1}{N-1}\\left[\\frac{\\text{B}\\left(\\stackrel{-}{x}\\right) - B\\left(\\underset{\\_}{x}\\right)}{\\text{c}\\left(\\stackrel{-}{x} - \\underset{\\_}{ x}\\right)}-1\\right]\u0026lt;1\\)\u003c/span\u003e\u003c/span\u003e\u003c/p\u003e \u003cp\u003elet \u003cspan class=\"InlineEquation\"\u003e\u003cspan class=\"mathinline\"\u003e\\(\\frac{\\text{B}\\left(\\stackrel{-}{x}\\right) - B\\left(\\underset{\\_}{x}\\right)}{\\text{c}\\left(\\stackrel{-}{x} - \\underset{\\_}{x}\\right)}=M,\\)\u003c/span\u003e\u003c/span\u003ewhere\u003cspan class=\"InlineEquation\"\u003e\u003cspan class=\"mathinline\"\u003e\\(1\u0026lt;M\u0026lt;N\\)\u003c/span\u003e\u003c/span\u003e\u003c/p\u003e \u003cp\u003e \u003cspan class=\"InlineEquation\"\u003e \u003cspan class=\"mathinline\"\u003e\\({\\delta }\\ge\\)\u003c/span\u003e \u003c/span\u003e \u003cspan class=\"InlineEquation\"\u003e\u003cspan class=\"mathinline\"\u003e\\(\\widehat{\\delta }\\equiv \\frac{M - 1}{N - 1}\u0026lt;1\\)\u003c/span\u003e\u003c/span\u003e\u003c/p\u003e \u003cp\u003eThis expression indicates in order to sustain cooperation, less extraction effort \u003cspan class=\"InlineEquation\"\u003e\u003cspan class=\"mathinline\"\u003e\\(x=\\underset{\\_}{x}\\)\u003c/span\u003e\u003c/span\u003emust be a SPE and the discount factor needs to be sufficiently high. For \u003cspan class=\"InlineEquation\"\u003e\u003cspan class=\"mathinline\"\u003e\\({\\delta }\u0026lt;\\)\u003c/span\u003e\u003c/span\u003e \u003cspan class=\"InlineEquation\"\u003e\u003cspan class=\"mathinline\"\u003e\\(\\widehat{\\delta }\\)\u003c/span\u003e\u003c/span\u003e(a critical discount factor), because of the free riding incentive, the business as usual\u003cspan class=\"InlineEquation\"\u003e\u003cspan class=\"mathinline\"\u003e\\(x=\\stackrel{-}{x}\\)\u003c/span\u003e\u003c/span\u003e will be an equilibrium extraction effort. However, if individuals are patient or very far sighted they will not easily deviate from the cooperation and temptation to free ride could be small. In addition to having large mental horizon, cooperation can be sustained with very little initial gain from deviation, large loss from counter threat or with short detection time for compliance after the game starts.\u003c/p\u003e \u003cp\u003eHowever, where resource users are involved in a repeated game and trigger strategies are used to sustain cooperation, individuals may cooperate and choose an effort\u003cspan class=\"InlineEquation\"\u003e\u003cspan class=\"mathinline\"\u003e\\(x=\\underset{\\_}{x}\u0026lt; \\stackrel{-}{x}\\)\u003c/span\u003e\u003c/span\u003e. The trigger strategy specifies that each party conforms to the agreement in each stage \u003cem\u003et\u003c/em\u003e as long as the parties have complied in all \u003cspan class=\"InlineEquation\"\u003e\u003cspan class=\"mathinline\"\u003e\\(t-1\\)\u003c/span\u003e\u003c/span\u003e prior stages; otherwise the parties deviate by breaking the arrangement.\u003c/p\u003e \u003cp\u003eProof for proposition \u003cspan refid=\"FPar3\" class=\"InternalRef\"\u003e2\u003c/span\u003e(ii):\u003cdiv id=\"Equl\" class=\"Equation\"\u003e\u003cdiv format=\"TEX\" class=\"mathdisplay\" id=\"FileID_Equl\" name=\"EquationSource\"\u003e\n$$\\frac{\\partial {V}_{i}^{c} - { V}_{i}^{d})}{\\partial N}=\\frac{\\partial }{\\partial N}\\left\\{\\frac{B\\left(\\underset{\\_}{x}\\right) - cN\\underset{\\_}{x}}{(1 - \\delta )}\\right\\}-\\frac{\\partial }{\\partial N}\\left\\{\\text{B}\\left(\\stackrel{-}{x}\\right)-c\\stackrel{-}{x}-c\\left(N-1\\right)\\underset{\\_}{x}+\\frac{\\delta [\\text{B}\\left(\\stackrel{-}{x}\\right) - cN\\stackrel{-}{x})] }{\\left(1 - \\delta \\right)}\\right\\}$$\u003c/div\u003e\u003c/div\u003e\u003cdiv id=\"Equm\" class=\"Equation\"\u003e\u003cdiv format=\"TEX\" class=\"mathdisplay\" id=\"FileID_Equm\" name=\"EquationSource\"\u003e\n$$=c\\underset{\\_}{x}- \\frac{c\\underset{\\_}{x}}{1 - \\delta }+\\frac{\\delta c\\stackrel{-}{x}}{1 - \\delta }=c\\underset{\\_}{x}-\\frac{c\\underset{\\_}{x}}{1 - \\delta }+\\frac{\\delta c\\stackrel{-}{x}}{1 - \\delta }$$\u003c/div\u003e\u003c/div\u003e\u003c/p\u003e \u003cp\u003e \u003cspan class=\"InlineEquation\"\u003e \u003cspan class=\"mathinline\"\u003e\\(=\\)\u003c/span\u003e \u003c/span\u003e \u003cspan class=\"InlineEquation\"\u003e\u003cspan class=\"mathinline\"\u003e\\(\\frac{\\delta }{1 - \\delta }c\\stackrel{-}{x}-\\frac{\\delta }{1 - \\delta }c\\underset{\\_}{x}\\)\u003c/span\u003e\u003c/span\u003e\u003c/p\u003e \u003cp\u003e \u003cspan class=\"InlineEquation\"\u003e \u003cspan class=\"mathinline\"\u003e\\(\\frac{\\partial {V}_{i}^{c} -{ V}_{i}^{d})}{\\partial N}\\)\u003c/span\u003e \u003c/span\u003e \u003cspan class=\"InlineEquation\"\u003e\u003cspan class=\"mathinline\"\u003e\\(=\\left(\\stackrel{-}{x}-\\underset{\\_}{x}\\right)\\frac{\\delta }{1 - \\delta }c\u0026gt;0\\)\u003c/span\u003e\u003c/span\u003e ; For \u003cspan class=\"InlineEquation\"\u003e\u003cspan class=\"mathinline\"\u003e\\(0\u0026lt;\\delta \u0026lt;1\\)\u003c/span\u003e\u003c/span\u003e\u003c/p\u003e \u003cp\u003eThis indicates as long as group members are able to monitor sufficiently each other, motivating cooperation will be easier if the group size is relatively large. On the contrary, cooperation hardly sustains when number of individuals in the group is relatively few (i.e., \u003cem\u003eN\u003c/em\u003e is small), or if individuals are heavily discount the value of cooperating in the future (i.e., \u0026#120575; is small). Other studies for e.g., [8⦌ concluded that informal sanctions like social pressure and disapproval of decisions may increase cooperation for public good game.\u003c/p\u003e \u003cp\u003eAlthough in this paper the compliance constraint is limited to the condition in which only one of the individuals in the group deviate from the agreed level, the result can be generalized to the condition where \u003cem\u003es\u0026thinsp;\u0026lt;\u0026thinsp;N\u003c/em\u003e, individuals deviate from cooperation and as a result the game revert to business as usual then after.\u003c/p\u003e \u003cp\u003eA simple common pool resource game has used in this manuscript, but this may open for extended future research along these lines.\u003c/p\u003e \u003c/div\u003e"},{"header":"4. Summary","content":"\u003cp\u003eThis paper presents both the finitely game (in its one time version) and infinitely repeated dynamic game formulations in which \u003cem\u003eN\u003c/em\u003e members of a users\u0026rsquo; group apply extraction effort that maximizes payoff from a common pool resource. For the finitely repeated move game, individual users are assumed to be homogeneous. And the payoff-functions are assumed to be concave and symmetric. Hence unconstrained optimizations are applied to find equilibrium solutions easily. The one time game of the finitely repeated framework clearly indicates the individual unregulated (business as usual) harvesting effort exceeds the social welfare maximizing level. Typically, this indicates the existence of negative externality in an open access common pool resources.\u003c/p\u003e \u003cp\u003eAn expression for infinitely repeated resource extraction game, where players have only two strategies (low and high extraction efforts) and played prisoners\u0026rsquo; dilemma as an individual payoff maximizing at equilibrium has also been analyzed. This formulation entails users may form a users\u0026rsquo; group and design an arrangement that enables them to self-select cooperation over the business as usual equilibrium that happens in the Prisoner\u0026rsquo;s dilemma game. In this formulation of the game, in order for low extraction effort sustain as equilibrium, the business as usual equilibrium \u003cspan class=\"InlineEquation\"\u003e\u003cspan class=\"mathinline\"\u003e\\(x=\\stackrel{-}{x}\\)\u003c/span\u003e\u003c/span\u003e is used as a threat (trigger strategy) for any deviations from cooperation. Besides, the strategies chosen by each player are assumed to be observable.\u003c/p\u003e \u003cp\u003eHence, this study has revealed, for the infinitely repeated game formulation, the socially optimal extraction effort may sustain through a self-enforcing agreement provided that individuals are sufficiently patient. In other words, cooperation may sustain over a long period of time if the initial gain from deviation is small or losses from counter threat after deviation is large. Meanwhile, initial gain from deviation becomes small if the time of detection for compliance after the game starts is short and the number individuals in the group are relatively large. And losses from counter threat (reverting in to business as usual equilibrium) is large, if the group size is large, or if individuals are very patient (i.e, the common discount factor, \u0026#120575; is large).\u003c/p\u003e"},{"header":"Declarations","content":"\u003cp\u003e\u003cstrong\u003eEthical approval\u003c/strong\u003e\u003c/p\u003e\n\u003cp\u003eNot applicable\u003c/p\u003e\n\u003cp\u003e\u003cstrong\u003eCompeting interests\u003c/strong\u003e\u003c/p\u003e\n\u003cp\u003eI have assured that there is no potential financial or non-financial competing interests from others related with this manuscript.\u003c/p\u003e\n\u003cp\u003e\u003cstrong\u003eAuthors\u0026rsquo; contribution\u003c/strong\u003e\u003c/p\u003e\n\u003cp\u003eThe corresponding author (HTB) has carried out the study beginning from problem identification, design and writing the manuscript titled\u0026nbsp;\u003cem\u003ethe commons dilemma: Strategic common pool resource extraction behavior.\u0026nbsp;\u003c/em\u003eBesides, HTB has approved for publication in the\u0026nbsp;Environmental Modeling and Assessment.\u003c/p\u003e\n\u003cp\u003e\u003cstrong\u003eFunding\u003c/strong\u003e\u003c/p\u003e\n\u003cp\u003eNo funds or other support were received during the preparation of this manuscript.\u003c/p\u003e\n\u003cp\u003e\u003cstrong\u003eAvailability of data and materials\u003c/strong\u003e\u003c/p\u003e\n\u003cp\u003eThe materials used for this study can be provided on request.\u003c/p\u003e"},{"header":"References","content":"\u003col\u003e\n \u003cli\u003eDell\u0026rsquo;Angelo, J., D\u0026rsquo;odorico, P., Rulli, M. C., \u0026amp; Marchand, P. (2017). The tragedy of the grabbed commons: Coercion and dispossession in the global land rush. \u003cem\u003eWorld Development\u003c/em\u003e, \u003cem\u003e92\u003c/em\u003e, 1-12.\u003c/li\u003e\n \u003cli\u003eOstrom, E., Gardner, R., Walker, J., \u0026amp; Walker, J. (1994). \u003cem\u003eRules, games, and common-pool resources\u003c/em\u003e. University of Michigan press.\u003c/li\u003e\n \u003cli\u003eKamijo, Y., Nihonsugi, T., Takeuchi, A., \u0026amp; Funaki, Y. (2014). Sustaining cooperation in social dilemmas: Comparison of centralized punishment institutions. \u003cem\u003eGames and Economic Behavior\u003c/em\u003e, \u003cem\u003e84\u003c/em\u003e, 180-195.\u003c/li\u003e\n \u003cli\u003eAnderson, C. M., \u0026amp; Putterman, L. (2006). Do non-strategic sanctions obey the law of demand? The demand for punishment in the voluntary contribution mechanism. \u003cem\u003eGames and Economic Behavior\u003c/em\u003e, \u003cem\u003e54\u003c/em\u003e(1), 1-24.\u003c/li\u003e\n \u003cli\u003eCarpenter, J. P. (2007). Punishing free-riders: How group size affects mutual monitoring and the provision of public goods. Games and Economic Behavior, 60(1), 31-51.\u003c/li\u003e\n \u003cli\u003eFehr, E., \u0026amp; G\u0026auml;chter, S. (2000). Cooperation and punishment in public goods experiments. \u003cem\u003eAmerican Economic Review\u003c/em\u003e, \u003cem\u003e90\u003c/em\u003e(4), 980-994.\u003c/li\u003e\n \u003cli\u003eNikiforakis, N. (2008). Punishment and counter-punishment in public good games: Can we really govern ourselves?. \u003cem\u003eJournal of Public Economics\u003c/em\u003e, \u003cem\u003e92\u003c/em\u003e(1-2), 91-112.\u003c/li\u003e\n \u003cli\u003eNoussair, C., \u0026amp; Tucker, S. (2005). Combining monetary and social sanctions to promote cooperation. \u003cem\u003eEconomic Inquiry\u003c/em\u003e, \u003cem\u003e43\u003c/em\u003e(3), 649-660.\u003c/li\u003e\n \u003cli\u003eSwope, K. J. (2002). An experimental investigation of excludable public goods. \u003cem\u003eExperimental Economics\u003c/em\u003e, \u003cem\u003e5\u003c/em\u003e, 209-222.\u003c/li\u003e\n \u003cli\u003eOstrom, E., Walker, J., \u0026amp; Gardner, R. (1992). Covenants with and without a sword: Self-governance is possible. American political science Review, 86(2), 404-417.\u003c/li\u003e\n \u003cli\u003eKandori, M. (2006). \u003cem\u003eRepeated Games, Entry in The New Palgrave Dictionary of Economics\u003c/em\u003e (No. CIRJE-F-395). CIRJE, Faculty of Economics, University of Tokyo.\u003c/li\u003e\n \u003cli\u003eBarrett, S. (2005). The theory of international environmental agreements, Sais, Johns Hopkins University, Rome Building, 1619 Massachusetts Avenue, NW Washington, DC 20036-2213, USA.\u003c/li\u003e\n\u003c/ol\u003e"}],"fulltextSource":"","fullText":"","funders":[],"hasAdminPriorityOnWorkflow":false,"hasManuscriptDocX":true,"hasOptedInToPreprint":true,"hasPassedJournalQc":"","hasAnyPriority":false,"hideJournal":true,"highlight":"","institution":"","isAcceptedByJournal":false,"isAuthorSuppliedPdf":false,"isDeskRejected":"","isHiddenFromSearch":false,"isInQc":false,"isInWorkflow":false,"isPdf":false,"isPdfUpToDate":true,"isWithdrawnOrRetracted":false,"journal":{"display":true,"email":"[email protected]","identity":"researchsquare","isNatureJournal":false,"hasQc":true,"allowDirectSubmit":true,"externalIdentity":"","sideBox":"","snPcode":"","submissionUrl":"/submission","title":"Research Square","twitterHandle":"researchsquare","acdcEnabled":true,"dfaEnabled":false,"editorialSystem":"","reportingPortfolio":"","inReviewEnabled":false,"inReviewRevisionsEnabled":true},"keywords":"Common pool resource, Game, Extraction effort, Self-enforcing cooperation","lastPublishedDoi":"10.21203/rs.3.rs-2654790/v1","lastPublishedDoiUrl":"https://doi.org/10.21203/rs.3.rs-2654790/v1","license":{"name":"CC BY 4.0","url":"https://creativecommons.org/licenses/by/4.0/"},"manuscriptAbstract":"\u003cp\u003eThe paper presents the strategic decision of common pool resource users at local level. Common pool resources typically characterized by the difficulty to exclude appropriation by others and the high level of rivalry (subtract ability). In this study, an application of game theory has been adopted to show individual resource use behavior. A game that better captures this underlying mechanism is the common pool resource game. Common pool resources with unlimited access are generally characterized by the existence of negative externality. Assuming a symmetric users\u0026rsquo; payoff function, static analysis for a simple mathematical model using solution method for strategic common pool game come up with extraction effort levels exceed the socially optimal level at equilibrium. This implies that unregulated common pool resource game poses social dilemma; when players interact only once or finitely repeated. On the contrary, the infinitely repeated framework shows cooperation (socially optimal level of extraction) may sustain by a self-enforcing agreement provided individuals are sufficiently patient. This frame work indicates that cooperation may sustain for a long period of time, if the initial gain from deviation is small or losses from counter threat after deviation is large.\u003c/p\u003e","manuscriptTitle":"The commons dilemma: Strategic common pool resource extraction behavior","msid":"","msnumber":"","nonDraftVersions":[{"code":1,"date":"2023-03-15 14:38:41","doi":"10.21203/rs.3.rs-2654790/v1","editorialEvents":[{"type":"communityComments","content":0}],"status":"published","journal":{"display":true,"email":"[email protected]","identity":"researchsquare","isNatureJournal":false,"hasQc":true,"allowDirectSubmit":true,"externalIdentity":"","sideBox":"","snPcode":"","submissionUrl":"/submission","title":"Research Square","twitterHandle":"researchsquare","acdcEnabled":true,"dfaEnabled":false,"editorialSystem":"","reportingPortfolio":"","inReviewEnabled":false,"inReviewRevisionsEnabled":true}}],"origin":"","ownerIdentity":"88f83d9a-c2d4-442f-bbf4-dcbc4cdb0f65","owner":[],"postedDate":"March 15th, 2023","published":true,"recentEditorialEvents":[],"rejectedJournal":[],"revision":"","amendment":"","status":"posted","subjectAreas":[],"tags":[],"updatedAt":"2023-04-19T04:29:21+00:00","versionOfRecord":[],"versionCreatedAt":"2023-03-15 14:38:41","video":"","vorDoi":"","vorDoiUrl":"","workflowStages":[]},"version":"v1","identity":"rs-2654790","journalConfig":"researchsquare"},"__N_SSP":true},"page":"/article/[identity]/[[...version]]","query":{"redirect":"/article/rs-2654790","identity":"rs-2654790","version":["v1"]},"buildId":"7rjqhiLT3MXkJMwkYKINL","isFallback":false,"isExperimentalCompile":false,"dynamicIds":[84888],"gssp":true,"scriptLoader":[]}